By Werner Fenchel, Jakob Nielsen, Asmus L. Schmidt
This booklet by means of Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and intricate historical past. In 1938-39, Nielsen gave a chain of lectures on
discontinuous teams of motions within the non-euclidean airplane, and this led him - in the course of
World conflict II - to write down the 1st chapters of the ebook (in German). whilst Fenchel,
who needed to get away from Denmark to Sweden due to the German profession,
returned in 1945, Nielsen initiated a collaboration with him on what grew to become recognized
as the Fenchel-Nielsen manuscript. at the moment they have been either on the Technical
University in Copenhagen. the 1st draft of the Fenchel-Nielsen manuscript (now
in English) was once entire in 1948 and it used to be deliberate to be released within the Princeton
Mathematical sequence. in spite of the fact that, because of the speedy improvement of the topic, they felt
that immense adjustments needed to be made earlier than booklet.
When Nielsen moved to Copenhagen collage in 1951 (where he stayed till
1955), he used to be a lot concerned with the overseas association UNESCO, and the
further writing of the manuscript was once left to Fenchel. The information of Fenchel now
deposited and catalogued on the division of arithmetic at Copenhagen Univer-
sity comprise unique manuscripts: a partial manuscript (manuscript zero) in Ger-
man containing Chapters I-II (
I -15), and a whole manuscript (manuscript I) in
English containing Chapters I-V (
1-27). The records additionally comprise a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes particular reviews to Fenchel's writings of Chapters III-V. Fenchel,
who succeeded N. E. Nf/Jrlund at Copenhagen college in 1956 (and stayed there
until 1974), used to be a great deal concerned with a radical revision of the curriculum in al-
gebra and geometry, and centred his examine within the thought of convexity, heading
the overseas Colloquium on Convexity in Copenhagen 1965. for nearly twenty years
he additionally placed a lot attempt into his activity as editor of the newly began magazine Mathematica
Scandinavica. a lot to his dissatisfaction, this job left him little time to complete the
Fenchel-Nielsen venture the way in which he desired to.
After his retirement from the college, Fenchel - assisted through Christian Sieben-
eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - discovered time to
finish the booklet effortless Geometry in Hyperbolic house, which used to be released via
Walter de Gruyter in 1989 almost immediately after his demise. at the same time, and with a similar
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, removal a number of the vague issues that have been within the unique
manuscript. Fenchel advised me that he meditated removal components of the introductory
Chapter I within the manuscript, due to the fact that this could be coated by way of the ebook pointed out above;
but to make the Fenchel-Nielsen booklet self-contained he finally selected to not do
so. He did choose to pass over
27, entitled Thefundamental staff.
As editor, i began in 1990, with the consent of the criminal heirs of Fenchel and
Nielsen, to provide a TEX-version from the newly typewritten model (manuscript 2).
I am thankful to Dita Andersen and Lise Fuldby-Olsen in my division for hav-
ing performed an excellent task of typing this manuscript in AMS- TEX. i've got additionally had
much aid from my colleague J0rn B0rling Olsson (himself a scholar of Kate Fenchel
at Aarhus collage) with the facts studying of the TEX-manuscript (manuscript three)
against manuscript 2 in addition to with a common dialogue of the variation to the fashion
of TEX. In so much respects we made up our minds to stick to Fenchel's intentions. besides the fact that, turning
the typewritten variation of the manuscript into TEX helped us to make sure that the notation,
and the spelling of yes key-words, will be uniform through the booklet. additionally,
we have indicated the start and finish of an evidence within the ordinary variety of TEX.
With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and
to my nice aid and delight they agreed to submit the manuscript of their sequence
Studies in arithmetic. i'm such a lot thankful for this optimistic and speedy response. One
particular challenge with the book grew to become out to be the copy of the numerous
figures that are a vital part of the presentation. Christian Siebeneicher had at
first agreed to carry those in ultimate digital shape, yet by means of 1997 it turned transparent that he
would now not manage to locate the time to take action. even though, the writer provided an answer
whereby I should still bring certain drawings of the figures (Fenchel didn't go away such
for Chapters IV and V), after which they'd arrange the construction of the figures in
electronic shape. i'm very thankful to Marcin Adamski, Warsaw, Poland, for his high-quality
collaboration in regards to the real construction of the figures.
My colleague Bent Fuglede, who has personaHy recognized either authors, has kindly
written a brief biography of the 2 of them and their mathematical achievements,
and which additionally areas the Fenchel-Nielsen manuscript in its right standpoint. In
this connection i want to thank The Royal Danish Academy of Sciences and
Letters for permitting us to incorporate during this publication reproductions of images of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a couple of specific symbols, an inventory of notation with brief
explanations and connection with the particular definition within the e-book has been integrated. additionally,
a accomplished index has been extra. In either circumstances, all references are to sections,
not pages.
We thought of including a whole checklist of references, yet determined opposed to it because of
the overwhelming variety of examine papers during this sector. in its place, a miles shorter
list of monographs and different finished money owed correct to the topic has been
collected.
My ultimate and so much honest thank you visit Dr. Manfred Karbe from Walter de Gruyter
for his commitment and perseverance in bringing this e-book into life.
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Discontinuous Groups of Isometries in the Hyperbolic Plane
This e-book via Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and intricate historical past. In 1938-39, Nielsen gave a chain of lectures on
discontinuous teams of motions within the non-euclidean airplane, and this led him - in the course of
World battle II - to write down the 1st chapters of the e-book (in German). whilst Fenchel,
who needed to get away from Denmark to Sweden end result of the German profession,
returned in 1945, Nielsen initiated a collaboration with him on what grew to become recognized
as the Fenchel-Nielsen manuscript. at the moment they have been either on the Technical
University in Copenhagen. the 1st draft of the Fenchel-Nielsen manuscript (now
in English) was once entire in 1948 and it was once deliberate to be released within the Princeton
Mathematical sequence. in spite of the fact that, a result of fast improvement of the topic, they felt
that giant alterations needed to be made earlier than ebook.
When Nielsen moved to Copenhagen college in 1951 (where he stayed until eventually
1955), he was once a lot concerned with the overseas association UNESCO, and the
further writing of the manuscript used to be left to Fenchel. The documents of Fenchel now
deposited and catalogued on the division of arithmetic at Copenhagen Univer-
sity include unique manuscripts: a partial manuscript (manuscript zero) in Ger-
man containing Chapters I-II (
I -15), and a whole manuscript (manuscript I) in
English containing Chapters I-V (
1-27). The information additionally include a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes certain reviews to Fenchel's writings of Chapters III-V. Fenchel,
who succeeded N. E. Nf/Jrlund at Copenhagen collage in 1956 (and stayed there
until 1974), was once greatly concerned with a radical revision of the curriculum in al-
gebra and geometry, and targeted his learn within the thought of convexity, heading
the overseas Colloquium on Convexity in Copenhagen 1965. for nearly twenty years
he additionally placed a lot attempt into his task as editor of the newly begun magazine Mathematica
Scandinavica. a lot to his dissatisfaction, this job left him little time to complete the
Fenchel-Nielsen venture the way in which he desired to.
After his retirement from the collage, Fenchel - assisted via Christian Sieben-
eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - came across time to
finish the e-book straight forward Geometry in Hyperbolic area, which used to be released via
Walter de Gruyter in 1989 almost immediately after his dying. concurrently, and with an identical
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, removal some of the imprecise issues that have been within the unique
manuscript. Fenchel instructed me that he reflected elimination elements of the introductory
Chapter I within the manuscript, considering this could be coated by way of the e-book pointed out above;
but to make the Fenchel-Nielsen ebook self-contained he finally selected to not do
so. He did choose to pass over
27, entitled Thefundamental workforce.
As editor, i began in 1990, with the consent of the criminal heirs of Fenchel and
Nielsen, to provide a TEX-version from the newly typewritten model (manuscript 2).
I am thankful to Dita Andersen and Lise Fuldby-Olsen in my division for hav-
ing performed an excellent activity of typing this manuscript in AMS- TEX. i've got additionally had
much support from my colleague J0rn B0rling Olsson (himself a pupil of Kate Fenchel
at Aarhus collage) with the evidence analyzing of the TEX-manuscript (manuscript three)
against manuscript 2 in addition to with a normal dialogue of the difference to the fashion
of TEX. In so much respects we determined to stick with Fenchel's intentions. even though, turning
the typewritten version of the manuscript into TEX helped us to make sure that the notation,
and the spelling of convinced key-words, will be uniform in the course of the booklet. additionally,
we have indicated the start and finish of an explanation within the ordinary variety of TEX.
With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and
to my nice reduction and pride they agreed to put up the manuscript of their sequence
Studies in arithmetic. i'm such a lot thankful for this optimistic and speedy response. One
particular challenge with the ebook became out to be the replica of the various
figures that are an essential component of the presentation. Christian Siebeneicher had at
first agreed to carry those in ultimate digital shape, yet by way of 1997 it grew to become transparent that he
would now not be capable to locate the time to take action. notwithstanding, the writer provided an answer
whereby I may still carry targeted drawings of the figures (Fenchel didn't depart such
for Chapters IV and V), after which they might manage the construction of the figures in
electronic shape. i'm very thankful to Marcin Adamski, Warsaw, Poland, for his tremendous
collaboration about the real construction of the figures.
My colleague Bent Fuglede, who has personaHy recognized either authors, has kindly
written a quick biography of the 2 of them and their mathematical achievements,
and which additionally locations the Fenchel-Nielsen manuscript in its right viewpoint. In
this connection i want to thank The Royal Danish Academy of Sciences and
Letters for permitting us to incorporate during this publication reproductions of photos of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a couple of precise symbols, a listing of notation with brief
explanations and connection with the particular definition within the e-book has been incorporated. additionally,
a accomplished index has been additional. In either circumstances, all references are to sections,
not pages.
We thought of including a whole checklist of references, yet determined opposed to it as a result of
the overwhelming variety of examine papers during this sector. in its place, a miles shorter
list of monographs and different entire bills proper to the topic has been
collected.
My ultimate and so much honest thank you visit Dr. Manfred Karbe from Walter de Gruyter
for his commitment and perseverance in bringing this ebook into life.
Statistics on Special Manifolds
This publication is anxious with statistical research at the particular manifolds, the Stiefel manifold and the Grassmann manifold, taken care of as statistical pattern areas along with matrices. the previous is represented via the set of m x okay matrices whose columns are together orthogonal k-variate vectors of unit size, and the latter through the set of m x m orthogonal projection matrices idempotent of rank okay.
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Extra info for Discontinuous Groups of Isometries in the Hyperbolic Plane
Example text
A differential w E Ai is said to be closed if dw = O. Differentials of the form dw (for some w E Ai) are said to be exact or cohomologically trivial. ) Every exact form is closed. Hence the exact i-forms constitute a complex vector subspace of the space of closed i-forms. The corresponding quotient space is denoted by HbR(S) and called the de Rham cohomology group. Example 1. A function f E AD is closed if and only if it is constant. Therefore H~R(S) '::::' C. On the other hand, f is 8-closed, that is, 8f = 0, if and only if it is holomorphic (see the Example in Sect.
N}. Thus to the generators Ci of the group 7r( 2) The function F is algebraic over the field M(82 ) and it can be regarded as an n-valued function on 8 2 • Its values form the points of a surface 81, which is therefore called the Riemann surface of the algebraic function F. 1 C 8 2 be a discrete subset which contains the poles of all the functions C1, ... ,en and also the points p E 82 where the polynomial has multiple roots. The last points are the zeros of the discriminant of P. The submanifold is a Riemann surface. The connectedness of U is a nontrivial fact, which follows from the converse to Proposition l.